Representing Data and Comparing Bar Charts

Turning a data table into a graph

A graph makes patterns in a data set easier to see, but the type of graph must suit the data. Categories such as travel methods can be compared using bars. Measurements recorded over time can be shown as a time series. Paired measurements, such as temperature and sales on the same day, can be shown on a scatter chart.

Before drawing, identify what each value represents. You may need to count observations to obtain frequencies, add components to obtain totals, or calculate percentages. Give the graph a clear title, label its axes and include units where appropriate. Numerical scales should use equal distances for equal increases in value.

Drawing and reading a bar chart

A bar chart uses the heights of separate bars to represent frequencies or other stated quantities. It is suitable for categorical data and discrete numerical data. The bars have equal widths and gaps between them; the widths do not represent the values.

Suppose two classes record how their pupils travel to school:

To draw a bar chart for Class A, put the travel methods on the horizontal axis and frequency, in pupils, on the vertical axis. An evenly spaced scale from 0 to 20 would fit the data. Draw bars with heights 20, 10 and 10.

The tallest bar identifies the most common category: walking. Adding the frequencies gives the class total. When comparing bars, state the difference in context: Class A has 10 more walkers than cyclists.

Dual and multiple bar charts

A dual bar chart places two bars side by side for each category. For the travel table, each category would have one bar for Class A and one for Class B. A key identifies the classes, and both use the same frequency scale.

This arrangement makes category-by-category comparisons easy. Class B has 10 more walkers than Class A, while the numbers travelling by bus and cycling are equal.

A multiple bar chart extends this arrangement to more than two data sets. Adding a third class gives three bars in each category. Keep the order of the classes consistent within every group and use a clear key.

Composite and percentage bar charts

A composite bar chart, also called a stacked bar chart, puts the components of a total into one bar. For Class A, stack sections of heights 20, 10 and 10, giving a total height of 40. For Class B, the total height is 50.

Read a section’s size from the difference between its upper and lower boundaries. If Class A’s bus section runs from 20 to 30, it represents $30-20=10$ pupils, not 30 pupils. Composite bars let you compare both group totals and their components.

A percentage bar chart instead makes every bar reach 100%. Calculate each section using

$$\text{Percentage}=\frac{\text{component frequency}}{\text{group total}}\times100.$$

For Class A, walking accounts for $20\div40\times100=50\%$, while bus travel and cycling each account for 25%. For Class B, the corresponding percentages are 60%, 20% and 20%.

These two stacked forms answer different questions. Composite bars show that Class B is larger. Percentage bars show that walking makes up a greater share of Class B. Both classes have 10 bus passengers, but these represent 25% of Class A and only 20% of Class B. Equal counts need not mean equal proportions.

When combining groups, calculate a percentage from the combined frequencies and combined total. Here, 50 of the 90 pupils walk, so approximately 55.6% walk overall. Simply averaging the two class percentages would give each class equal weight even though their sizes differ.

Line charts for discrete numerical data

A vertical line chart represents frequencies using thin vertical lines rather than bars. Put the numerical outcomes on the horizontal axis and frequency on the vertical axis. Each line rises from zero to its outcome’s frequency.

For example, suppose 3 pupils have no pets, 8 have one pet, 5 have two pets and 2 have three pets. Draw separate vertical lines at 0, 1, 2 and 3 pets, with heights 3, 8, 5 and 2. The tallest line shows that the modal number of pets is 1. Adding the line heights gives 18 pupils altogether.

The horizontal positions must respect the numerical scale: an outcome of 4 belongs twice as far from 0 as an outcome of 2. Do not join the tops of these separate frequency lines.

Time series: following change over time

A time-series graph is a line graph with time on the horizontal axis and the recorded quantity on the vertical axis. Plot observations in chronological order and join successive points with straight lines. Unlike a vertical line chart, the connecting lines help show change from one time to the next.

Suppose a shop sells 40, 55, 45, 60, 50 and 70 items in six successive months.

Data for Sales over six months

The overall trend is upward: sales rise from 40 items in month 1 to 70 in month 6. However, there are short-term falls from month 2 to month 3 and from month 4 to month 5. A useful description distinguishes the general trend from these fluctuations rather than claiming that sales increase every month.

When comparing two time series, compare their levels, overall directions and fluctuations over the same period. Check the scales before judging how steep or variable either series looks.

Scatter charts: keeping measurements paired

A scatter chart displays bivariate data: two numerical measurements for each observation. Each pair becomes one point. Where one variable is used to explain or predict the other, put that explanatory variable on the horizontal axis and the response variable on the vertical axis.

Suppose a shop records these pairs of temperature and ice-cream sales on five days: $(12,20)$, $(15,28)$, $(18,25)$, $(21,40)$ and . Temperature is measured in °C and sales are numbers of ice creams. The first point belongs at 12°C horizontally and 20 ice creams vertically; its two values must stay together.

Data for Temperature and ice-cream sales

Plot separate crosses and do not join them. There is no time order implied by moving from one point to the next.

To interpret the chart, look at the overall pattern. Points tending upwards from left to right show a positive relationship; points tending downwards show a negative relationship. If there is no clear pattern, there is no clear relationship visible. In this example, higher temperatures tend to accompany higher sales, although the points do not all lie on a straight line.

A relationship on a scatter chart does not by itself prove that one variable causes the other. Other factors may also affect the measurements.